The present invention relates to an interleaving A/D conversion type waveform digitizer module and a test apparatus thereof. More particularly, the present invention relates to a digitizer module for detecting measurement errors accompanied by phase errors of sampling timing in interleaving A/D converting and correcting the errors.
An N way interleaving. A/D (i.e. analog/digital) conversion type digitizer module, which uses N A/D converters, is capable of increasing a sampling rate in appearance, but on the other hand it is required to sample signals with a precise sampling timing.
In this example, it is described below to let a interleaving constant be 2. And, it is described to let the number of time series data be 4096 pieces of data, which are 2 to the power 12. The digitizer module includes two A/D converters and a Fourier-transform unit. The A/D converters convert analog signals into digital signals at a constant sampling rate. The two A/D converters increase the sampling rate in appearance by sampling the analog signals in turn. In this example, it is to let the data sampled by the two A/D converters be 4096 data series. The Fourier-transform unit performs Fourier-transform of the digital signals sampled by the A/D converters.
The Fourier-transform unit receives the digital signal data series sampled by the A/D converters and outputs 4096 pieces of frequency spectrum data resulting from a fast Fourier-transform (FFT). The Fourier-transform unit includes a first FFT unit, a second FFT unit and a butterfly operation unit. Each of the first FFT unit and the second FFT unit receives 2048 time series data and outputs 2048 pieces of mid-data (complex data) resulting from FFT. The butterfly operation unit performs the last part of a butterfly operation well known as used for FFT process.
The butterfly operation unit performs a butterfly operation on the data from the first FFT unit and the second FFT unit and outputs 4096 pieces of frequency spectrum data resulting from the well-known butterfly operation to which FFT process is applied.
As a configuration example of a digitizer module for the semiconductor type test apparatus, there is a digitizer module including a first A/D converter to which analog signals are transmitted by a device to be tested, a second A/D converter, an arrangement unit and a Fourier-transform unit. Here, the two A/D converters have the entirely same characteristic of sampling timing including a group delay characteristic and an aperture delay characteristic. Usually, sampling data sampled by the two A/D converters is stored first in a buffer memory installed, then the data is provided to the Fourier-transform unit and an operation is performed thereon.
Analog signals for testing outputted by a device to be tested are provided to input terminals of both the first A/D converter and the second A/D converters and the first A/D converter performs sampling on even data series. It is to let outputted even time series data be D0, D2, D4, . . . . And, the second A/D converter performs sampling on odd data series. It is to let outputted odd time series data be D1, D3, D5, . . . . The arrangement unit receives the two time series data and outputs time series data D0, D1, D2, D3, D4, D5, . . . arranged in turn.
Phase adjustment should be done to make phase differences between the sampling timings of the two A/D converters be a constant difference. And also if there is a phase error, proper frequency spectrum data cannot be obtained because FFT process is performed on data sampled with a constant time interval.
As described above, in prior art, sampling timing of a plurality of A/D converters does not change and a sampling clock rate is constant or within an error-allowed range. Meanwhile, a sampling characteristic of the A/D converter brings about a change in sampling with a constant time interval, which is desirable, because of a difference in quality of parts of A/D converter itself, surroundings temperature, a change according to time laps and a change in power source voltage. And, the group delay characteristic is changed according to a change in a clock frequency for sampling in the case of using, for example, the semiconductor type test apparatus substantially changing the sampling frequency. Accompanied by these causes, a change arises from sampling timing of an ideal state. That is, in the case of desiring a high precision frequency spectrum of an input signal, a practical problem not preferable to the prior apparatus.
Therefore, it is an object of the present invention to provide an interleaving A/D conversion type digitizer module and a semiconductor test apparatus for correcting an operation process of a Fourier-transform unit on the basis of phase difference between the sampling timings of a plurality of A/D converters. The above and other objects can be achieved by combinations described in the independent claims. The dependent claims define further advantageous and exemplary combinations of the present invention.
According to the first aspect of the present invention, a digitizer module for converting an analog signal outputted by an electronic device into a digital signal, comprises at least two A/D converters for converting an analog signal outputted by the electronic device into a digital signal, each of the A/D converters operating with different sampling timing, a Fourier-transform unit for performing Fourier-transform on each of the digital signals converted by the at least two A/D converters and outputting a transformed signal, and an interleaving unit for generating a data sequence in which the transformed signal outputted by the Fourier-transform unit is comprised, wherein the interleaving unit comprises a spurious elimination means for eliminating a spurious component in the transformed signal resulting from a phase error between an ideal sampling timing, with which each of the at least two A/D converters should perform sampling on each of the analog signals, and the sampling timing, with which each of the at least two A/D converters performs sampling on each of the analog signals.
The interleaving unit further may comprise an aliasing elimination means for eliminating an aliasing component of the spurious component.
The spurious elimination means may calculate a correction factor on the basis of each of the transformed signals by using the phase error and a boundary condition, under which the spurious component in the transformed signal does not exist, and eliminates the spurious component on the basis of the correction factor.
The spurious elimination means may calculate the correction factor for each of the transformed signals outputted by the Fourier-transform unit corresponding to each of the at least two A/D converters, and eliminates the spurious component on the basis of the correction factor.
The ideal sampling timing may be defined as a sampling timing with which each of other A/D converters performs sampling in turn at a constant time interval in case one of sampling timings of the at least two A/D converters is regarded as a reference sampling timing, and the spurious elimination means may calculate the correction factor on the basis of each of the phase errors between each of sampling timings of the other A/D converters and the ideal sampling timing, and may eliminate the spurious component on the basis of the correction factor.
The spurious elimination means may multiply each of the transformed signals by the correction factor calculated for each of the transformed signals.
The spurious elimination means may calculate the correction factor to eliminate components other than a signal component of the analog signal and the aliasing component of the signal component when a total sum of the N transformed signals multiplied by the correction factor are calculated.
The spurious elimination means may calculate the correction factor for each of a plurality of bands, into which a band of the transformed signal is divided, on the basis of a phase of sampling timing of the at least two A/D converters.
The spurious elimination means may calculate the correction factor by using a simultaneous equation.
An sampling pulse, with which the at least two A/D converters perform sampling on the analog signal, is given by:                     p        m            ⁡              (        t        )              =                  ∑                  r          =                      -            ∞                          ∞            ⁢              σ        ⁢                  {                                    (                              t                -                                                      (                                                                  N                        xc3x97                        r                                            +                      m                                        )                                    ⁢                  Ts                                            )                        -                          τ              m                                }                      ,
where N denotes the number of the at least two A/D converters, m denotes an integer in a range of 0(zero) to (N-1), t denotes time, Ts denotes a phase interval of each of the at least two A/D converters, m denotes an m-th A/D converter andxcfx84 denotes the phase error of the at least two A/D converters, a Fourier-transform of sampling series of the analog signal sampled by the at least two A/D converters is given by:                     Xm        ⁡                  (          f          )                    =                        1                      N            xc3x97            Ts                          ⁢                              ∑                          r              =                              -                ∞                                      ∞                    ⁢                                    X              ⁡                              (                                  f                  -                                      r                                          N                      xc3x97                      Ts                                                                      )                                      ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  r                  2                                ⁢                                  (                                      m                    +                                                                  τ                        ⁢                                                  xe2x80x83                                                ⁢                        m                                            Ts                                                        )                                                                          ,          
        ⁢          that      ⁢              xe2x80x83            ⁢      is        ,          "IndentingNewLine"        ⁢                            X          0                ⁡                  (          f          )                    =                                    x            _                    ⁡                      (                          -              k                        )                          +        …        +                              x            _                    ⁡                      (                          -              1                        )                          +                              x            _                    ⁡                      (            0            )                          +                              x            _                    ⁡                      (            1            )                          +        …        +                              x            _                    ⁡                      (            l            )                                                  X        1            ⁡              (        f        )              =                            c          1                      -            k                          ⁢                              x            _                    ⁡                      (                          -              k                        )                              +      …      +                        c          1                      -            1                          ⁢                              x            _                    ⁡                      (                          -              1                        )                              +                        x          _                ⁡                  (          0          )                    +                        c          1                ⁢                              x            _                    ⁡                      (            1            )                              +      …      +                        c          1          l                ⁢                              x            _                    ⁡                      (            l            )                                                  X        2            ⁡              (        f        )              =                            c          2                      -            k                          ⁢                              x            _                    ⁡                      (                          -              k                        )                              +      …      +                        c          2                      -            1                          ⁢                              x            _                    ⁡                      (                          -              1                        )                              +                        x          _                ⁡                  (          0          )                    +                        c          2                ⁢                              x            _                    ⁡                      (            1            )                              +      …      +                        c          2          l                ⁢                              x            _                    ⁡                      (            l            )                                ⋮                    X                  N          -          1                    ⁡              (        f        )              =                            c                      N            -            1                                -            k                          ⁢                              x            _                    ⁡                      (                          -              k                        )                              +      …      +                        c                      N            -            1                                -            1                          ⁢                              x            _                    ⁡                      (                          -              1                        )                              +                        x          _                ⁡                  (          0          )                    +                        c                      N            -            1                          ⁢                              x            _                    ⁡                      (            1            )                              +      …      +                        c                      N            -            1                    l                ⁢                              x            _                    ⁡                      (            l            )                              
(where, in case a band of X(f) is [xe2x88x922 fs, 2 fs], terms having r in a range of xe2x88x92k to 1 in the above equation are components within a band [0, 4 fs], and are cm and x(r) are respectively given by:             c      m        =          ⅇ                        -                                    j              ⁢                              xe2x80x83                            ⁢              π                        2                          ⁢                  (                      r            +                                          τ                m                            /              Ts                                )                      ,                    x        _            ⁡              (        r        )              =                  1        NTs            ⁢              X        ⁡                  (                      f            -                          r              NTs                                )                      ,
and
in case the aliasing component related to 2 fs, which is a frequency of signal component x{circumflex over ( )}(0), is x{circumflex over ( )}(u) (where x{circumflex over ( )} is a substitute notation for {overscore (x)}.), the spurious elimination means may calculate the correction factor L1, L2, . . . LN-1 to satisfy a equation given by:
X0(f)+L1X1(f)+L2X2(f)+. . . +LN-1XN-1(f)=a{overscore (x)}(0)+b{overscore (x)}(u), 
where either a or b is a random real number.
The Fourier-transform unit outputs the transformed signal DFTm(r) resulting from a Fourier-transform of the digital signal outputted by the at least two A/D converters, and for a first band, in which the signal component x{circumflex over ( )}(0) exists, the spurious elimination means calculates a Fourier-transform X(f)=X(r/NTs) of the analog signal using a equation given by:                               X          ⁡                      (                          r              NTs                        )                          =                ⁢                              1                          1              +                              L                1                            +                              L                2                            +              …              +                              L                                  N                  -                  1                                                              ⁢                      {                                                            DFT                  0                                ⁡                                  (                  r                  )                                            +                                                L                  1                                ⁢                                  ⅇ                                                            -                      j                                        ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                                          xe2x80x83                                        ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                                          r                      N                                        ⁢                                          (                                              1                        +                                                                              τ                            1                                                    /                          Ts                                                                    )                                                                      ⁢                                                      DFT                    1                                    ⁡                                      (                    r                    )                                                              +                                                                      ⁢                                            L              2                        ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  r                  N                                ⁢                                  (                                      2                    +                                                                  τ                        2                                            /                      Ts                                                        )                                                      ⁢                                          DFT                2                            ⁡                              (                r                )                                      ⁢                          xe2x80x83                        ⁢            …                    +                                                            ⁢                                    L                              N                -                1                                      ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  r                  N                                ⁢                                  (                                      N                    -                    1                    +                                                                  τ                                                  N                          -                          1                                                                    /                      Ts                                                        )                                                      ⁢                                          DFT                                  N                  -                  1                                            ⁡                              (                r                )                                              }                ,            
while for a second band, in which aliasing component x{circumflex over ( )}(u) of the signal component x{circumflex over ( )}(0) exists, the spurious elimination means may calculate a Fourier-transform X(f)=X(r/NTs) of the analog signal using a equation given by:                               X          ⁡                      (                          r              NTs                        )                          =                ⁢                              1                          1              +                                                ∑                                      n                    =                    1                                                        N                    -                    1                                                  ⁢                                                      c                    n                    u                                    ⁢                                      L                    n                                                                                ⁢                      {                                                            DFT                  0                                ⁡                                  (                  r                  )                                            +                                                L                  1                                ⁢                                  ⅇ                                                            -                      j                                        ⁢                                          xe2x80x83                                        ⁢                    2                    ⁢                                          xe2x80x83                                        ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                                          r                      N                                        ⁢                                          (                                              1                        +                                                                              τ                            1                                                    /                          Ts                                                                    )                                                                      ⁢                                                      DFT                    1                                    ⁡                                      (                    r                    )                                                              +                                                                      ⁢                                            L              2                        ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  r                  N                                ⁢                                  (                                      2                    +                                                                  τ                        2                                            /                      Ts                                                        )                                                      ⁢                                          DFT                2                            ⁡                              (                r                )                                      ⁢                          xe2x80x83                        ⁢            …                    +                                                            ⁢                                    L                              N                -                1                                      ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  r                  N                                ⁢                                  (                                      N                    -                    1                    +                                                                  τ                                                  N                          -                          1                                                                    /                      Ts                                                        )                                                      ⁢                                          DFT                                  N                  -                  1                                            ⁡                              (                r                )                                              }                .            
The first band may be in a frequency range of 0 to 2 fs, and the second band may be in a frequency range of 2 fs to frequency 4 fs.
A digitizer module comprising four A/D converters, wherein a sampling pulse with which the four A/D converters perform sampling on the analog signal is given by:                     p        m            ⁡              (        t        )              =                  ∑                  r          =                      -            ∞                          ∞            ⁢              σ        ⁢                  {                                    (                              t                -                                                      (                                                                  4                        ⁢                        r                                            +                      m                                        )                                    ⁢                  Ts                                            )                        -                          τ              m                                }                      ,
where m denotes an integer 0 to 3, t denotes time, Ts denotes a phase interval of each of the four A/D converters, m denotes an m-th A/D converter andxcfx84 denotes the phase error of each of the four A/D converters, a Fourier-transform of sampling series of the analog signal sampled by each of the four A/D converters is given by:                                           Xm            ⁡                          (              f              )                                =                                    1                              4                ⁢                Ts                                      ⁢                                          ∑                                  r                  =                                      -                    ∞                                                  ∞                            ⁢                                                X                  ⁡                                      (                                          f                      -                                              r                                                  4                          ⁢                          Ts                                                                                      )                                                  ⁢                                  ⅇ                                                            -                      j                                        ⁢                                          xe2x80x83                                        ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                                          r                      2                                        ⁢                                          (                                              m                        +                                                                              τ                            ⁢                                                          xe2x80x83                                                        ⁢                            m                                                    Ts                                                                    )                                                                                                          ,                                          that          ⁢                      xe2x80x83                    ⁢          is                ,                                                      X            0                    ⁡                      (            f            )                          =                                            x              _                        ⁡                          (                              -                1                            )                                +                                    x              _                        ⁡                          (              0              )                                +                                    x              _                        ⁡                          (              1              )                                +          …          +                                    x              _                        ⁡                          (              5              )                                                                                    X            1                    ⁡                      (            f            )                          =                                            c              1                              -                1                                      ⁢                                          x                _                            ⁡                              (                                  -                  1                                )                                              +                                    x              _                        ⁡                          (              0              )                                +                                    c              1                        ⁢                                          x                _                            ⁡                              (                1                )                                              +          …          +                                    c              1              5                        ⁢                                          x                _                            ⁡                              (                5                )                                                                                                  X            2                    ⁡                      (            f            )                          =                                            c              2                              -                1                                      ⁢                                          x                _                            ⁡                              (                                  -                  1                                )                                              +                                    x              _                        ⁡                          (              0              )                                +                                    c              2                        ⁢                                          x                _                            ⁡                              (                1                )                                              +          …          +                                    c              2              5                        ⁢                                          x                _                            ⁡                              (                5                )                                                                                                                X              3                        ⁡                          (              f              )                                =                                                    c                3                                  -                  1                                            ⁢                                                x                  _                                ⁡                                  (                                      -                    1                                    )                                                      +                                          x                _                            ⁡                              (                0                )                                      +                                          c                3                            ⁢                                                x                  _                                ⁡                                  (                  1                  )                                                      +            …            +                                          c                3                5                            ⁢                                                x                  _                                ⁡                                  (                  5                  )                                                                    ,            
(where, in case a band of X(f) is [xe2x88x922 fs, 2 fs], terms having r in a range of xe2x88x921 to 5 in the above equation are components within a band [0, 4 fs], and cm and x(r) are respectively given by:                               c          m                =                  ⅇ                                    -                                                j                  ⁢                                      xe2x80x83                                    ⁢                  π                                2                                      ⁢                          (                              r                +                                                      τ                    m                                    /                  Ts                                            )                                          ,                                    x            _                    ⁡                      (            r            )                          =                              1                          4              ⁢              Ts                                ⁢                      X            (                          f              -                              r                                  4                  ⁢                  Ts                                                      )                                )    ,
and
in case the aliasing component related to 2 fs, which is a frequency of signal component x{circumflex over ( )}(0), is x{circumflex over ( )}(4) (where x{circumflex over ( )} is a substitute notation for {overscore (x)}), the spurious elimination means may calculate the correction factor L1, L2 and L3 to satisfy a equation given by:
X0(f)+L1X1(f)+L2X2(f)+. . . +L3X3(f)=a{overscore (x)}(0)+b{overscore (x)}(4), 
where either a or b is a random real number.
For a third band in a frequency range of 0 to fs, the correction factor L1, L2 and L3 may be represented as:                               L          1                =                  -                                                                      c                  1                                ⁡                                  (                                                            c                      2                                        -                    1                                    )                                            ⁢                              (                                                      c                    3                                    -                  1                                )                            ⁢                              (                                                                            c                      2                                        ⁢                                          c                      3                                                        +                                      c                    2                                    +                                      c                    3                                                  )                                                                    (                                                      c                    1                                    -                                      c                    2                                                  )                            ⁢                              (                                                      c                    1                                    -                                      c                    3                                                  )                            ⁢                              (                                                                            c                      1                                        ⁢                                          c                      2                                                        +                                                            c                      2                                        ⁢                                          c                      3                                                        +                                                            c                      3                                        ⁢                                          c                      1                                                                      )                                                                                      L          2                =                  -                                                                      c                  2                                ⁡                                  (                                                            c                      3                                        -                    1                                    )                                            ⁢                              (                                                      c                    1                                    -                  1                                )                            ⁢                              (                                                                            c                      3                                        ⁢                                          c                      1                                                        +                                      c                    3                                    +                                      c                    1                                                  )                                                                    (                                                      c                    2                                    -                                      c                    3                                                  )                            ⁢                              (                                                      c                    2                                    -                                      c                    1                                                  )                            ⁢                              (                                                                            c                      2                                        ⁢                                          c                      3                                                        +                                                            c                      3                                        ⁢                                          c                      1                                                        +                                                            c                      1                                        ⁢                                          c                      2                                                                      )                                                                                                  L            3                    =                      -                                                                                c                    3                                    ⁡                                      (                                                                  c                        1                                            -                      1                                        )                                                  ⁢                                  (                                                            c                      2                                        -                    1                                    )                                ⁢                                  (                                                                                    c                        1                                            ⁢                                              c                        2                                                              +                                          c                      1                                        +                                          c                      2                                                        )                                                                              (                                                            c                      3                                        -                                          c                      1                                                        )                                ⁢                                  (                                                            c                      3                                        -                                          c                      2                                                        )                                ⁢                                  (                                                                                    c                        3                                            ⁢                                              c                        1                                                              +                                                                  c                        1                                            ⁢                                              c                        2                                                              +                                                                  c                        2                                            ⁢                                              c                        3                                                                              )                                                                    ,            
for a fourth band in a frequency range of fs to 2 fs, the correction factor L1, L2 and L3 mat be represented as:                               L          1                =                  -                                                    (                                                      c                    2                                    -                  1                                )                            ⁢                              (                                                      c                    3                                    -                  1                                )                                                                                      c                  1                                ⁡                                  (                                                            c                      1                                        -                                          c                      2                                                        )                                            ⁢                              (                                                      c                    1                                    -                                      c                    3                                                  )                                                                                      L          2                =                  -                                                    (                                                      c                    3                                    -                  1                                )                            ⁢                              (                                                      c                    1                                    -                  1                                )                                                                                      c                  2                                ⁡                                  (                                                            c                      2                                        -                                          c                      3                                                        )                                            ⁢                              (                                                      c                    2                                    -                                      c                    1                                                  )                                                                                                  L            3                    =                      -                                                            (                                                            c                      1                                        -                    1                                    )                                ⁢                                  (                                                            c                      2                                        -                    1                                    )                                                                                                  c                    3                                    ⁡                                      (                                                                  c                        3                                            -                                              c                        1                                                              )                                                  ⁢                                  (                                                            c                      3                                        -                                          c                      2                                                        )                                                                    ,            
for a fifth band in a frequency range of 2 fs to 3 fs, the correction factor L1, L2 and L3 may be represented as:                               L          1                =                  -                                                    (                                                      c                    2                                    -                  1                                )                            ⁢                              (                                                      c                    3                                    -                  1                                )                                                                                      c                  1                                ⁡                                  (                                                            c                      1                                        -                                          c                      2                                                        )                                            ⁢                              (                                                      c                    1                                    -                                      c                    3                                                  )                                                                                      L          2                =                  -                                                    (                                                      c                    3                                    -                  1                                )                            ⁢                              (                                                      c                    1                                    -                  1                                )                                                                                      c                  2                                ⁡                                  (                                                            c                      2                                        -                                          c                      3                                                        )                                            ⁢                              (                                                      c                    2                                    -                                      c                    1                                                  )                                                                                      L          3                =                  -                                                    (                                                      c                    1                                    -                  1                                )                            ⁢                              (                                                      c                    2                                    -                  1                                )                                                                                      c                  3                                ⁡                                  (                                                            c                      3                                        -                                          c                      1                                                        )                                            ⁢                              (                                                      c                    3                                    -                                      c                    2                                                  )                                                        
and for a sixth band in a frequency range of 3 fs to 4 fs, the correction factor L1, L2 and L3 may be represented as:                               L          1                =                  -                                                    (                                                      c                    2                                    -                  1                                )                            ⁢                              (                                                      c                    3                                    -                  1                                )                            ⁢                              (                                  1                  +                                      c                    2                                    +                                      c                    3                                                  )                                                                                      c                  1                  2                                ⁡                                  (                                                            c                      1                                        -                                          c                      2                                                        )                                            ⁢                              (                                                      c                    1                                    -                                      c                    3                                                  )                            ⁢                              (                                                      c                    1                                    +                                      c                    2                                    +                                      c                    3                                                  )                                                                                      L          2                =                  -                                                    (                                                      c                    3                                    -                  1                                )                            ⁢                              (                                                      c                    1                                    -                  1                                )                            ⁢                              (                                  1                  +                                      c                    3                                    +                                      c                    1                                                  )                                                                                      c                  2                  2                                ⁡                                  (                                                            c                      2                                        -                                          c                      3                                                        )                                            ⁢                              (                                                      c                    2                                    -                                      c                    1                                                  )                            ⁢                              (                                                      c                    1                                    +                                      c                    2                                    +                                      c                    3                                                  )                                                                                      L          3                =                  -                                                    (                                                      c                    1                                    -                  1                                )                            ⁢                              (                                                      c                    2                                    -                  1                                )                            ⁢                              (                                  1                  +                                      c                    1                                    +                                      c                    2                                                  )                                                                                      c                  3                  2                                ⁡                                  (                                                            c                      3                                        -                                          c                      1                                                        )                                            ⁢                              (                                                      c                    3                                    -                                      c                    2                                                  )                            ⁢                              (                                                      c                    1                                    +                                      c                    2                                    +                                      c                    3                                                  )                                                        
The Fourier-transform unit outputs the transformed signal DFTm(r) resulting from a Fourier-transform of the digital signal outputted by the at least two A/D converters, and the spurious elimination means may calculate a Fourier-transform X(f)=X(r/NTs) of the analog signal for the third and fourth bands using a equation given by:             X      ⁡              (                  r          NTs                )              =                  1                  1          +                      L            1                    +                      L            2                    +                      L            3                              ⁢              {                                            DFT              0                        ⁡                          (              r              )                                +                                    L              1                        ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                π                ⁢                                  r                  N                                ⁢                                  (                                      1                    +                                                                  τ                        1                                            /                      Ts                                                        )                                                      ⁢                                          DFT                1                            ⁡                              (                r                )                                              +                                    L              2                        ⁢                          ⅇ                                                -                  j2π                                ⁢                                  r                  N                                ⁢                                  (                                      2                    +                                                                  τ                        2                                            /                      Ts                                                        )                                                      ⁢                                          DFT                2                            ⁡                              (                r                )                                              +                                    L              3                        ⁢                          ⅇ                                                -                  j2π                                ⁢                                  r                  N                                ⁢                                  (                                      3                    +                                                                  τ                        3                                            /                      Ts                                                        )                                                      ⁢                                          DFT                3                            ⁡                              (                r                )                                                    }              ,
while the spurious elimination means calculates a Fourier-transform X(f)=X(r/NTs) of the analog signal for the fifth and sixth bands using a equation given by:       X    ⁡          (              r        NTs            )        =            1              1        +                              ∑                          n              =              1                        3                    ⁢                                    c              n              4                        ⁢                          L              n                                            ⁢                  {                                            DFT              0                        ⁡                          (              r              )                                +                                    L              1                        ⁢                          ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                π                ⁢                                  r                  N                                ⁢                                  (                                      1                    +                                                                  τ                        1                                            /                      Ts                                                        )                                                      ⁢                                          DFT                1                            ⁡                              (                r                )                                              +                                    L              2                        ⁢                          ⅇ                                                -                  j2π                                ⁢                                  r                  N                                ⁢                                  (                                      2                    +                                                                  τ                        2                                            /                      Ts                                                        )                                                      ⁢                                          DFT                2                            ⁡                              (                r                )                                              +                                    L              3                        ⁢                          ⅇ                                                -                  j2π                                ⁢                                  r                  N                                ⁢                                  (                                      3                    +                                                                  τ                        3                                            /                      Ts                                                        )                                                      ⁢                                          DFT                3                            ⁡                              (                r                )                                                    }            .      
According to the second aspect of the present invention, a test apparatus for testing an electronic device, comprises a pattern generator for generating a pattern signal and a expectation signal, a waveform adjuster for shaping a waveform of the pattern signal generated by the pattern generator, a device contacting unit for providing the pattern signal shaped by the waveform adjuster to the electronic device installed, and receiving analog signal outputted by the electronic device, a digitizer module for converting the analog signal outputted by the electronic device into digital signal and a decision unit for deciding quality of the electronic device by comparing the expectation signals outputted by the pattern generator with signals outputted by the digitizer module, wherein the digitizer module comprises at least two A/D converters for converting an analog signal outputted by the electronic device into a digital signal, each of the A/D converters operating with different sampling timing, a Fourier-transform unit for performing Fourier-transform on each of the digital signals converted by the at least two A/D converters, and outputting a transformed signal and an interleaving unit for generating a data sequence in which the transformed signal outputted by the Fourier-transform unit is comprised, and the interleaving unit comprises a spurious elimination means for eliminating a spurious component in the transformed signal resulting from a phase error between an ideal sampling timing, with which each of the at least two A/D converters should perform sampling on each of the analog signals, and the sampling timing, with which each of the at least two A/D converters performs sampling on each of the analog signals.
The summary of the invention does not necessarily describe all necessary features of the present invention. The present invention may also be a sub-combination of the features described above. The above and other features and advantages of the present invention will become more apparent from the following description of the embodiments taken in conjunction with the accompanying drawings.